DEEPAWALI ASSIGNMENT

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1 fo/u fopkjr Hkh# tu] ugha vkjehks dke] foifr ns[k NksM+s rqjar e/;e eu dj ';kea iq#"k flag ladyi dj] lgrs foifr vusd] ^cuk^ u NksM+s /;s; dks] j?kqcj jk[ks VsdAA jfpr% ekuo /kez iz.ksrk ln~xq# Jh j.knksm+nklth egkjkt Subject : MATHEMATICS Available Online : HOME ASSIGNMENT Objective: Vector, D & Complex Subjective: Misc. Topics R Wishing You & Your Family A Very Happy & Prosperous Deepawali DEEPAWALI ASSIGNMENT Address : Plot No. 7, III- Floor, Near Patidar Studio, Above Bond Classes, Zone-, M.P. NAGAR, Bhopal : , , WhatsApp

2 Question bank on Vector, D and Complex Number & Misc. Subjective Problem FREE Download Study Package from website: & Select the correct alternative : (Only one is correct) Q. /vec If a, b, a b, then a + b is : (A) (B*) (C) (D) Q. 5/vec The position vector of a point P moving in space is given by OP R (cos t)î + (cos t) ĵ+ (5sin t) kˆ. The time 't' when the point P crosses the plane x y + z 5 is [Hint: (A) π sec (B*) 6 π sec (C) π sec (D) π sec put x cos t ; y cos t ; z 5 sin t in the equation of the plane, we get cos t cos t + sin t 5 π sin t t sec ] 6 Q. 6/vec Indicate the correct order sequence in respect of the following : I. x y + 6 y + 6 x y z The lines and are orthogonal. II. The planes x y z and the plane x y z are orthogonal. III. The function f (x) ln(e + e x ) is monotonic increasing x R. IV. If g is the inverse of the function, f (x) ln(e + e x ) then g(x) ln(e x e ). (A) FTFF (B) TFTT (C*) FFTT (D) FTTT [Sol. I. L is to î ĵ kˆ V L is to î ĵ+ kˆ V V + L is not perpendicular to L False Q. /complex If [Hint: E V II. () ( ) ()( ) + + planes are not perpendicular False III. f (x) ln(e + e x ) f ' (x) e x x > f is increasing x R True e + e IV. y ln(e + e x ) e + e x e y e x e y e f (y) ln(e y e ) g (x) ln(e x e ) True] z z is purely imaginary then z + z z z is equal to : (A*) (B) (C) (D) + (z (z z ) z ) + xi xi ] Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phone : , page

3 Q.5 7/vec In a regular tetrahedron, the centres of the four faces are the vertices of a smaller tetrahedron. The ratio FREE Download Study Package from website: & of the volume of the smaller tetrahedron to that of the larger is n m, where m and n are relatively prime positive integers. The value of (m + n) is (A) (B) (C) 7 (D*) 8 [Hint: V l [a b c ] ; V 6 s [a b c ] 6 7 V s Hence m n m Vl 7 or k n 7 m and n are relatively prime k, (m + n) 8 further hint for a b c V s [a b c ] ] Q.6 9/vec If a, b, c are non-coplanar unit vectors such that a x( bx c) ( b+ c) then the angle between a & b is (A*) π/ (B) π/ (C) π/ (D) π [JEE 95, ] Q.7 /vec The sine of angle formed by the lateral face ADC and plane of the base ABC of the tetrahedron ABCD where A (,, ); B (,, 5); C (,, ) and D (,, ) is (A) 9 (B*) 5 9 (C) 9 Q.8 6/complex Let z be a complex number, then the region represented by the inequality z + < z + is given by : (A*) Re (z) > (B) Im (z) < (C) Re (z) < & Im (z) > (D) Re (z) < & Im (z) > Q.9 /vec The volume of the parallelpiped whose edges are represented by the vectors a i j + k, b i j + k, c i + j k is : (A*) 7 (B) 5 (C) (D) none Q. 5/vec Let u,v, w be the vectors such that u + v+ w, if u, v & w 5 then the value of u.v + v.w + w. u is : (A) 7 (B*) 5 (C) (D) 5 [JEE 95,] Q. 6/vec Let a î ĵ, b ĵ kˆ, c î (D) kˆ. If d is a unit vector such that a. d [ b, c, d] then d (A*) ± (î+ ĵ kˆ ) (B) ± (î+ ĵ kˆ ) (C) ± (î+ ĵ+ kˆ ) (D) ± 6 kˆ 9 Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phone : , page

4 Q. 8/complex If z be a complex number for which z [Hint : (A*) + (B) + (C) z z + z + z z z r r + r r Now consider all inequalities ] +, then the greatest value of z is : z (D) none Q. /vec If the non zero vectors a & b are perpendicular to each other, then the solution of the equation, r a b is : [Hint: (A*) r xa + ( a b) a. a b. b a b (C) r xa b (D) r x b a where x is any scalar. r x a + y b + a b take cross with a r a yb a + z(a b) a b y(b a) + z[(a a) b (b a) a] ] b y(b a) + z (a a) b since b < a b are non coplanar z (a a) & y z a r x a + (a b) a ] (B) r x b ( ) x y z x y z 5 Q. /vec The lines and are coplanar if k k (A) k or (B*) k or (C) k or (D) k or Q.5 /vec Which one of the following statement is INCORRECT? (A) If n. a, n. b & n. c for some non zero vector n, then[ a b c ] (B*) there exist a vector having direction angles α º & β 5º (C) locus of point for which x & y is a line parallel to the z - axis whose distance from the z-axis is 5 (D) In a regular tetrahedron OABC where ' O ' is the origin, the vector OA + OB + OC is perpendicular to the plane ABC. Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phone : , page

5 [Explanation: (A) n is perpendicular to a, b as well as c a b c a b c,, must be in the same plane [ ] (B) If one direction angle is θ then the remaining two cannot be less than 9 θ a + b + c (D) verify that a. ( a + b + c) where a b c ] Q.6 /complex Given that the equation, z + (p + iq) z + r + is has a real root where p, q, r, s R. Then which one is correct (A) pqr r + p s (B) prs q + r p (C) qrs p + s q (D*) pqs s + q r [Hint: Let z α be the real root α + (p + iq)α + r + is (α + pα + r) + i(qα + s) + i qα + s...() and α + pα + r...() From () α s. Put in () to get the result ] q Q.7 7/vec The distance of the point (,, 5) from x-axis is (A) (B) 5 (C) (D*) [Hint: distance from x-axis of x, y, z z y + ] Q.8 9/vec Given non zero vectors A, B and C, then which one of the following is false? (A) A vector orthogonal to A B and C is ± (A B) C (B) A vector orthogonal to A + B and A B is ± A B (C) Volume of the parallelopiped determined bya, B and C is A B C A B (D*) Vector projection ofa onto B is B [Hint: (A B) It should be B B ] /vec Given three vectors a, b, c such that they are non zero, non coplanar vectors, then which of the Q. 9 following are non coplanar. (A*) a + b, b + c, c + a (B) a b, b + c, c + a (C) a + b, b c, c + a (D) a + b, b + c, c a [Hint: Verify v + v v in order to quickly answer ] Q. 6/complex The sum i + i + i i, where i is equal to (A) i (B) + 999i (C) + i (D*) + i [Sol. S i + i + i i is i + i i + i S( i) i + i + i i i Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phone : , page 5

6 i( i i ) i + i + i i( + i) + i i + i ( + i )( + i) S i + i i + i ] Q. /vec Locus of the point P, for which OP + i represents a vector with direction cosine cos α ( ' O ' is the origin) is : (A) A circle parallel to y z plane with centre on the x axis (B*) a cone concentric with positive x axis having vertex at the origin and the slant height equal to the magnitude of the vector (C) a ray emanating from the origin and making an angle of 6º with x axis (D) a disc parallel to y z plane with centre on x axis & radius equal to O P sin 6º [Hint : ] Q. 8/vec A line with direction ratios (,, ) intersects the lines r ĵ+ λ(î + ĵ + kˆ ) r î + µ (î + ĵ + kˆ) at A and B, then l (AB) is equal to (A*) (B) (C) (D) x y + z x + y z [Hint: L : λ ; L : µ Hence any point on L and L can be (λ, λ, λ) and (µ, µ, µ) µ λ µ λ + µ λ solving µ and λ A (,, ) and B (,, ) ] Q. 7/vec The vertices of a triangle are A (,, ), B (,, ) & C (,, 5). The vector representing the internal bisector of the angle A is : (A) i + j + k (B) i j + k (C) i + j k (D*) i + j + k Q. 8/complex Lowest degree of a polynomial with rational coefficients if one of its root is, + i is (A) (B*) (C) 6 (D) 8 [Sol. Let x + i ( ) x x + x x + x x x 8x x x + 9 ] and Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phone : , page 6

7 Q.5 55/vec A plane vector has components & w.r.t. the rectangular cartesian system. This system is rotated FREE Download Study Package from website: & through an angle π (A), in anticlockwise sense. Then w.r.t. the new system the vector has components : (B*) [Hint : cos θ sin θ 5 5 now w.r.t. new system X Y X component is5 cos (θ 5º) Y component is5 sin (θ 5º) ] 7, (C) 7, (D) none Q.6 56/vec Let a xi + j k ; b i + xj + k & c i + k. If the ordered set [ b c a ] is left handed, then: (A) x (, ) (B) x (, ) (C*) x (, ) (D) x {, } [Sol. For a right hand set [ a b c ] > and for a left handed system [ a b c ] < ] Q.7 7/vec If cos α î + ĵ+ kˆ, î + cosβ ĵ+ kˆ & î + ĵ+ cos γ kˆ (α β γ n π) are coplanar then the value of [Hint : α β γ cosec + cosec + cosec equal to (A) (B*) (C) (D) none of these cosα cosβ cosγ sin α β sin β γ sin sin cos γ + sin α sin α cosα cosβ cosβ cos γ β + sin cos γ + sin cos γ γ + sin β sin γ β sin sin sin γ γ + + sin β γ sin multiply by cosec α cosec β cosec γ Alternatively : Expand number () cosec γ + cosec β + cosec α (cos α ) [ cos γ (cos β ) ( cos γ) ] + ( cos β) ( cos γ) or ( cos α) ( cos β) cos γ + ( cos β) ( cos γ) + ( cos γ) ( cos α) Now proceed ] Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phone : , page 7

8 Q.8 /complex The straight line ( + i)z + (i ) z i on the complex plane, has intercept on the imaginary axis equal to (A*) 5 (B) 5 [Hint: put z iy ( + i) iy (i ) i y i y + y y 5 ] (C) 5 (D) 5 Q.9 75/vec The perpendicular distance of an angular point of a cube of edge 'a' from the diagonal which does not pass that angular point, is [Sol. (A) a (B) a (C) a Consider a unit cube equation of AB is r kˆ + λ(î + ĵ kˆ ) p.v. of N λ, λ, ( λ) now Hence ON λî + λĵ ( + λ)kˆ ON. AB λ + λ + l + λ λ / ON î ĵ kˆ ; ON ] (D*) Q. 88/vec Which one of the following does not hold for the vector V a x ( b x a )? (A) perpendicular to a (B*) perpendicular to b (C) coplanar with a & b (D) perpendicular to a x b. Q. 5/complex Let z, z & z be the complex numbers representing the vertices of a triangle ABC respectively. If P is a point representing the complex number z satisfying ; a (z z ) + b (z z ) + c (z z ), then w.r.t. the triangle ABC, the point P is its : (A) centroid (B) orthocentre (C) circumcentre (D*) incentre [Hint : az + bz + cz z (a + b + c) z az + bz + cz a + b + c z is the incentre] Q. 96/vec Given the position vectors of the vertices of a triangle ABC, A (a) ; B (b) ; C (c). A vector r is parallel to the altitude drawn from the vertex A, making an obtuse angle with the positive Y-axis. If r ; a î ĵ kˆ ; b î + ĵ kˆ ; c î ĵ kˆ, then r is (A*) 6î 8ĵ 6kˆ (B) 6 î 8ĵ+ 6kˆ (C) 6 î 8ĵ+ 6kˆ (D) 6 î + 8ĵ+ 6kˆ a Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phone : , page 8

9 FREE Download Study Package from website: & [Sol. r Equation of line BC: î + ĵ kˆ + λ ( i j+ k ) p.v. of N is λ +, λ, λ vector now BC AN (λ ) î + ( λ) ĵ + (λ ) kˆ AN. BC (λ ) ( λ) + (λ ) (λ + 9λ + λ) λ /7 9 î + ĵ + 9kˆ AN 7 r P (AN) r P. AN P P ; 6 AN 7 or hence 7 r + 9î + ĵ+ 9kˆ 7 + ( î + ĵ+ kˆ ) angle with y axis is ve +ve sign to be rejected r 6î 8ĵ 6kˆ (A) ] 7 Q. 7/complex The complex number, z 5 + i has an argument which is nearly equal to : (A) π/ (B*) π/6 (C) π/ (D) π/8 [Hint : z 5 + i 5 + i 6 (cosθ + i sinθ) + + i 6(cosθ + i sinθ) i 676(cosθ + i sinθ) 676 sin θ 76 and 676 cos θ 8 θ π θ π 6 tan θ 76 8 ~ Q. 97/vec If a î + ĵ+ kˆ, b î ĵ+ kˆ, c î + ĵ kˆ, then the value of [Hint : [ ] ] P a.a b.a c.a a.b b.b c.b (A) (B) (C*) 6 (D) 6 a b c 6 ] a.c b.c equal to c.c Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phone : , page 9

10 Q.5 96/complex If the equation x + ax + b where a, b R has a non real root whose cube is then (7a + b) has the value (A*) 98 (B) 9 (C) 98 (D) 9 [Sol. the cube root of are the roots of x (x 7)(x + 7x + 9) where a 7 and b 9 7a + b 98 Ans.] Direction for Q.6 to Q.. Let A î + ĵ+ kˆ and B î + ĵ+ 5kˆ Q.6 (i)/vec The value of the scalar A B + (A B) is equal to (A) 8 (B) 7 (C*) 7 (D) 6 [Sol. a b Ans] Q.7 (ii)/vec Equation of a line passing through the point with position vector î + ĵ and orthogonal to the plane containing the vectors A and B, is (A*) r ( λ + )î (λ )ĵ + λkˆ (C) r λî + (λ ) ĵ λkˆ [Sol. A B î ĵ kˆ 5 (B) r ( λ )î (λ )ĵ + λkˆ (D) none ( )î (5 9)ĵ+ ( 6) kˆ î + ĵ kˆ (î ĵ+ kˆ ) Here r î + ĵ+ λ(î ĵ+ kˆ ) r ( + λ)î + ( λ) ĵ+ λkˆ Ans.] Q.8 (iii)/vec Equation of a plane containing the point with position vector ( î ĵ+ kˆ ) and parallel to the vectors A and B, is (A) x + y + z (B) x y z (C*) x y + z (D) x + y + z [Sol. n î ĵ+ kˆ a î ĵ+ kˆ ( r a) n r (î ĵ+ kˆ ) a n x y + z ] Q.9 (iv)/vec Volume of the tetrahedron whose coterminous edges are the vectors A, B and C î + ĵ kˆ is (A) (B*) / (C) 8/ (D) 8 [Sol. [a b c ] [( 6 5) ( ) + ( 8)] 6 [ + 5] 6 8 ] Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phone : , page

11 Q. (v)/vec Vector component of A perpendicular to the vector B is given by B (A B) A (A B) B (A B) A (A B) (A*) (B) (C) (D) B B A A A B [Sol. x A B B (A) ] Select the correct alternatives : (More than one are correct) Q. 5/vec If a, b, c are different real numbers and a i + b j + ck ; b i + c j + a k & ci + a j + b k are position vectors of three non-collinear points A, B & C then : (A*) centroid of triangle ABC is a + b + c ( i + j + k ) (B*) i + j + k is equally inclined to the three vectors (C*) perpendicular from the origin to the plane of triangle ABC meet at centroid (D*) triangle ABC is an equilateral triangle. Q. 5/vec The vectors a, b, c are of the same length & pairwise form equal angles. If a i + j & b j+ k, the pv's of c can be : (A*) (,, ) (B),, (C),, (D*),, [Hint : Let c x i + y j + z k x + y + z () now a. b b. c c. a I y + z x + y () z x y x put z and y in terms of x in () to get x and then get y and z ] Q. 5/complex Which of the following locii of z on the complex plane represents a pair of straight lines? (A*) Re z (B*) Im z (C) z + z (D) z z i [ Hint : C negative real axis ; D perpendicular bisector of the line joining (, ) & (, ) ] Q. 56/vec If a, b, c & d are linearly independent set of vectors & Ka + K b + K c + K d then : (A*) K + K + K + K (B*) K + K K + K (C*) K + K K + K (D) none of these [Hint : ka + kb + kc + kd a, b, c, d are linearly independent k k k k ] Q.5 57/vec If a, b, c & d are the pv's of the points A, B, C & D respectively in three dimensional space & satisfy the relation a b + c d, then : (A*) A, B, C & D are coplanar (B) the line joining the points B & D divides the line joining the point A & C in the ratio :. (C*) the line joining the points A & C divides the line joining the points B & D in the ratio : (D*) the four vectors a, b, c & d are linearly dependent. Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phone : , page

12 FREE Download Study Package from website: & [Hint : a + c b + d b + d Hence line joining A & C intersect line joining B & C ] Q.6 59/complex If z i z i z then z can be equal to : (A) i (B*) i (C*) + i (D*) i [Hint : (z i) (z i) z i or z i e i π/ z + i or i ] Q.7 59/vec If a & b are two non collinear unit vectors & a, b, x a y b form a triangle, then : (A*) x ; y & a + b a b cos (B*) x ; y & cos a b (C) a + b a b cot + a + b cos a, ( a + b) a b cos & x, y [Hint : a, b & xa y b form a triangle hence, a + b + xa y b (D) none (x + ) â + ( y) b Since & a b are collinear x & y Also a + b +. a â.( â+ bˆ ) Also cos φ â+ bˆ b ( + cos θ). a + b cos θ cos a b + cos θ a + b A (where φ is the angle between â & a + b ) a + b cos φ ( + cos θ) a + b cos φ + cos θ ] Q.8 5/vec The lines with vector equations are ; r ( ~ i j i + j + k ) r ( ~ i j i + j + k ) µ are such that : λ and (A) they are coplanar (B*) they do not intersect (C*) they are skew (D*) the angle between them is tan (/7) Q.9 5/complex Given a, b, x, y R then which of the following statement(s) hold good? (A*) (a + ib) (x + iy) a ib x + y (B*) ( ix) ( + ix) a ib a + b (C*) (a + ib) (a ib) x iy x + iy (D*) (y ix) (a + ib) y + ix a ib [Hint : Modulus of a complex number, which is the ratio of two conjugates is unity. Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phone : , page

13 e.g. in A, a + ib x + iy a ib a + ib a ib x + iy x + y ] Q.5 5/vec The acute angle that the vector i j + k makes with the plane contained by the two vectors [Hint : i + j k and i j + k is given by : (A) cos (B*) sin n a b i j k i j k (C) tan ( ) (D*) cot ( ) ( + ) ( + ) 5 ( i j + k ) i n j + k v i j + k v cos π θ sin θ v. n ] i j + k Q.5 58/vec The volume of a right triangular prism ABCA B C is equal to. If the position vectors of the vertices of the base ABC are A(,, ) ; B(,, ) and C(,, ) the position vectors of the vertex A can be: (A*) (,, ) (B) (,, ) (C) (,, ) (D*) (,, ) [Hint : knowing the volume of the prism we find its altitude H (A A ) A (x, y, z ) relate the co-ordinates of the vector AA AA (x, y, z ) and its length. We get the other equation from the contition perpendicular to AC ] OR compute ± AB AC AB AC n 6 n A A ± ( i + j + k ) (x ) î + (y ) ĵ + (z ) k Compare to get at the possible coordinates of A ] π Q.5 58/complex If x r CiS for r n r, n N then : (A*) Limit n Re (C) Limit n Im r n x r r n x r r (B) Limit n Re (D*) Limit n Im Q.5 5/vec If a line has a vector equation, r i + 6 j + ( i j ) holds good? n x r r 6 and designating the vertex n x r r λ then which of the following statements Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phone : , page

14 (A) the line is parallel to i + 6 j (B*) the line passes through the point i + j (C*) the line passes through the point i + 9 j (D*) the line is parallel to xy plane [Hint : Line is parallel to i j D Also put r i + j for which λ and r i + j for which λ B & C ] 9 Q.5 55/vec If a, b, c are non-zero, non-collinear vectors such that a vector π π a c b then p + q is [Sol. ( ) p a b cos ( a b ) c and a vector q a c cos ( ( )) (A) parallel to a (B*) perpendicular to a (C*) coplanar with b & c (D) coplanar with a and c p a b cos(π θ) c where θ is the angle betweena and b and q a ccos( π φ) b where φ is the angle betweena and c now p + q (a bcos ) c a c cos b θ φ (a.b) c (a.c) b a (c b) B and C ] Q.55 59/complex The greatest value of the modulus of of the complex number ' z ' satisfying the equality z + z is (A) + 5 SUBJECTIVE: (B*) + 5 (C) 5 Q. 9/5 Let a î ĵ and b î + ĵ and x a + (q ) b, y p a + qb. If x y, then express p as a function of q, say p f (q), (p & q ) and find the intervals of monotonicity of f (q). [Sol. ( ) (D*) q x î ĵ + (q ) î + ĵ î (q ) + ĵ ( ) ĵ + q î + y p î ĵ x y gives p q(q ) Ans. dp [q ] > dq q > 5 + Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phone : , page

15 q > or q < and decreasing in q (, ), q Ans. ] ln x e Q. 5/ Using only the limit theorems Lim and Lim. Evaluate Lim x x x x [Ans. ] [Sol. x x x Lim x ln x x + l x ln x ln x e e Lim x ln x x + Lim e ln x x ln x ln x [e ] x ln x ln x (x ln x ln x) ln x x x + ln x(x )(x ) () () Lim () () () Lim x (x )( ln x x + ) put x + h, as x, h h Lim h ln( + h) h put ln ( + h) y + h e y y (e ) Lim y y y (e ) e y l and Lim y Ans. ] y y x (x ) ln x x x + y e y Lim y Lim y y y () Lim y y e + y y e y x x x ln x x x + Q. 9/5 The three vectors a i j + k ; b i j k and c i + k are all drawn from the point with p.v. î ĵ. Find the equation of the plane containing their end point in scalar dot product form. i + j k. r ] [ Ans. ( ) π n n + Q. / (cos x cos x) dx zπ zπ n where n N [Sol. I (cos x) ( cos x) dx as f is even. Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phone : , page 5

16 zπ (cos x) n z n. sin x dx t. dt when cosx t. n + n+ L NM t O QP n + ] Q.5 97/5 Let points P, Q & R have position vectors, r i j k; r i + j + k & r i+j k respectively, relative to an origin O. Find the distance of P from the plane OQR. [Ans : units] [Sol. n r r î ĵ kˆ î ( 6 ) ĵ( 8) + kˆ ( 6) î + ĵ 5kˆ î ĵ+ kˆ nˆ d ( î ĵ+ kˆ) (î ĵ+ kˆ) Pr ojection of OPon n Q.6 8/ Evaluate: ( x )(x )(x ) dx [Ans. /] [Sol. I (x )( x)(x ) dx let π I π sin put θ t 6 + units] x cos θ + sin θ dx sin θ dθ x sin θ ; x cos θ and x cos θ + sin θ sin θ cos θ π sin θ cos θ cos θ sin θdθ sin θ cosθ dθ π π dt I / sin t cos t / (sin put sin t y t cos t)dt θ cos θ sin θ cos θ dθ Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phone : , page 6

17 FREE Download Study Package from website: & I y dy y Ans. ] Q.7 98/5 Given that vectors A, B, C form a triangle such that [Sol. A B+ C, find a,b,c,d such that the area of the triangle is5 6 where A ai + bj + ck; B di + j + k & C i + j k. [Ans: ( 8,,, ) or (8,,, 5)] [ REE 9, 6 ] A B + C ai + bj + ck ( d + ) i + ( + ) j + ( ) k ( d + ) i + j + k Hence d + a; b and c again B x C 5 6 B C ( B C) 5 (5 + d ) (d + 8) 5 (5 + d ) (d 5) 5 now proceed to get two values of d ] n Q.8 6/ n k k+ n k k + π Lim x x dx [Ans. ] n n n 8 k n β k k + [Sol. Let ( x α)( β x) dx where α ; β α n n x α cos α + β sin θ dx (β α) sin θ cos θ x α (β α)sin θ π I (β α) ( β α) (sin θcos θ) dθ sin put I θ t ( β α) π sin ( β α) π 8 t dt π ( β α) 8 π ( β α) sin π 8 n π θdθ t dt which is independent of k. Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phone : , page 7

18 n l Lim π n n k 8 n π n π π Lim () Lim n Ans. ] n n 8 k n 8n 8 Q.9 /5 Find the distance of the point P(i + j + k) from the plane L which passes through the three points A(i + j + k), B(i + j + k), C(i + j + k). Also find the pv of the foot of the perpendicular from P on the plane L. [Sol. a î ĵ+ kˆ b î ĵ+ kˆ a b î ĵ [ Ans :,,, kˆ î () ĵ( ) + kˆ( ) a b î + ĵ+ kˆ n (say) BP î ĵ+ kˆ c PN Projection of c on n c.n n + + Now equation of a line through P and n is r î + ĵ+ kˆ + λ (î + ĵ+ kˆ ) + λ [ î + ĵ+ kˆ ] Let the position vector of N (+λ), (+λ), (+λ) Now a, b and Q. Evaluate: (a) AN ( λ ) î + λĵ+ λkˆ AN must be copalnar λ λ λ [λ] + [λ + λ ] l λ / Position vector of N,, ] ] sin x + cos x π dx, x, ; (b) sin x cos x sin x + cos x dx sin x cos x Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phone : , page 8

19 [Sol.(a)I (b) sin x + cos x π dx, x, [ sin x cos x cos x + tan x cos x + tan x dx dx sin x cos x sin x put cot x t cot x cosec x dx dt + t I dt t put + t y t dt y dy I C I put y y y + dy dy t y + t t + l n where t cot x Ans(a). t + + sin x + cos x sin x + cot dx sin x cos x sin x cos x tan x t + t t dt + t + cot x cot x cosec x dx cot x dy dy y y + y C ln y + x dx + tan x tan x sec x dx tan x t + + l n + C, where t tan x Ans(b). ] t + + Q. 5/5 Find the equation of the straight line which passes through the point with position vector a, meets the line r b + tc and is parallel to the plane r.n. [Sol. Suppose the required line intersects the given line at P with p.v.( b + t c ). As the line l is to the plane r.n. Hence AP.n [( b a) + t c] n t (a b).n c.n Hence equation of the line is r a + λ (b a + tc) + λ + b).n (a r a b a c c.n + λ b).n (a r a (a b) c c.n Ans ] dx Q. Integrate:. [Ans. [tan (sin x + cos x) + cos x sin x ln + sin x + cos x sin x cos x + C] Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phone : , page 9

20 dx [Sol. I cos x sin x (cos x sin x)dx sin x) ( sin x)( + (cosx sin x)dx + dx (cos x sin x)dx (cos x sin x)( + sin x cos x) (cos x sin x) ( + sin x) I ( (sin x + cos x) )( + (sin x cos x) ) put sin x + cos x t (cos x sin x) dx dt dt ( t ) + ( + t ) dt dt hence I ( t )( + t ) dt ( t )( + t ) + + t t tan (t) + ln [tan (sin x + cos x) + + t + C t ln + sin x + cos x sin x cos x + C Ans. ] Q. 7/5 Find the equation of the line passing through the point (,, ) which is perpendicular to both of the lines x y + z x + y z + and Also find all points on this line the square of whose distance from (,, ) is 57. x y z [Ans., ; ( 9,, ) ; (,, ) ] 6 [Sol. Equation of the line passing through (,, ) x y z a b c since () is perpendicular to hence a + b + c and a + b c a b c () x y + z a x y z hence the equation of the lines is 6 now any point P on () can be taken as λ ; 6λ + ; λ + distance of P from Q (,, ) (λ) + (6λ) + λ 57 and b 6 x + c...() Ans. y z + Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phone : , page

21 ( )λ 57 λ or Hence Q is ( 9,, ) or (,, ) Ans.] Q. Lim n [Sol. L n + n n Lim n + n ( e n n + n n + n ) n [Ans. e ] e l, where l + + n n ( n) Lim n + n + n n n + + n n Lim Lim n + n (n + ) n n n (n + n) (n + ) n + n n n Lim n n + n + (n + ) (rationalisation) Lim n n + n + n + n + (n + ) n Lim Lim n n n n n n n n L e ans. ] Q.5 5/5 If z-axis be vertical, find the equation of the line of greatest slope through the point (,, ) on the plane x + y z. [Sol. Equation of the line of greatest slope x y + z a b c where a + b c...() now equation of the horizontal plane is z i.e. x + y + z now a vector along the line of intersection of given plane and horizontal plane is v î ĵ kˆ ( î ĵ) î + ĵ + kˆ since the line of greatest slope is also perpendicular to the vector v hence a + b + c...() from () and () a + b c a + b + c a + 8 b c + 9 equation of the line of greatest slope a 8 b c x y + 8 z ] Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phone : , page

22 FREE Download Study Package from website: & π Q.6 Let I [Sol. π cos x dx and J a cos x + bsin x Compute the values of I and J. π ai + bj...() π sin x dx, where a > and b >. a cos x + bsin x bcos x a sin x and bi aj dx a cos x + bsin x π b bi aj ln [ a cos x + bsin x] bi aj ln a from () and () aπ a I + abj b I abj b ln( b a) I a again abi + b I aπ b + b l n Ans. + b a bπ and abi a J a ln ( b a) subtract...() bπ b J a l n Ans. a + b a Alternatively: convert a cos x + b sin x into a single cosine say cos(x + f) and put x f t ] Teko Classes, Maths : Suhag R. Kariya (S. R. K. Sir), Bhopal Phone : , page